Changes

*A shift matrix U with ones on the superdiagonal is an upper shift matrix.

*A shift matrix U with ones on the superdiagonal is an upper shift matrix.

*The alternative subdiagonal matrix L is unsurprisingly known as a lower shift matrix.  

*The alternative subdiagonal matrix L is unsurprisingly known as a lower shift matrix.  

*Let Z is a shift matrix , then ZA, ZTA, AZ, AZT, ZAZT are equal to the matrix A shifted one position down, up left, right, and down along the main diagonal respectively.

*Let Z is a shift matrix , then <math>ZA, Z^TA, AZ, AZ^T, ZAZ^T</math> are equal to the matrix A shifted one position down, up left, right, and down along the main diagonal respectively.

*The alternative subdiagonal matrix L is unsurprisingly known as a lower shift matrix.  

*The alternative subdiagonal matrix L is unsurprisingly known as a lower shift matrix.  

*The (i,j):th component of U and L are

*The <math>(i,j)^th</math> component of U and L are

  U_{ij} = \delta_{i+1,j}, \quad L_{ij} = \delta_{i,j+1},where \delta_{ij} is the Kronecker delta symbol.

  <math>U_{ij} = \delta_{i+1,j}, \quad L_{ij} = \delta_{i,j+1}</math>,where <math>\delta_{ij}</math> is the Kronecker delta symbol.

*For example, the 5×5 shift matrices are:

*For example, the 5×5 shift matrices are:

*All shift matrices are nilpotent; an n by n shift matrix S becomes the null matrix when raised to the power of its dimension n.

*All shift matrices are nilpotent; an n by n shift matrix S becomes the null matrix when raised to the power of its dimension n.